Three years ago, Danica Patrick was in the midst of her motorsports career, unsure of when she might stop driving and uncertain what her life after sports might include. She thought she wanted a child. Someday. She still thinks that. So, she decided to give herself the best chance of having a child in her post-racing life.

She decided to freeze her eggs.

During one offseason, Patrick went through rigorous hormone treatment to produce, retrieve and freeze her then 33-year-old eggs. After two weeks of self-administered hormone injections, her doctor removed 24 eggs — 19 of which were mature. Her doctor told her she needed 20 eggs to all-but guarantee that one could someday become a healthy baby.

“This is an insurance policy,” Patrick says. “It doesn’t mean I can’t try a different way if I want to. I just wanted to have options.”

Patrick is now 36 years old. She retired after two final races this year — the Daytona 500 and the...